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1Ec170NewCournotHand(2)

1Ec170NewCournotHand(2) - Notes on the Cournot Model I...

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Notes on the Cournot Model I. Assumptions: 2 identical firms, linear demand, and constant marginal cost Market Demand is P = a-bQ (where Q is market output) and MC = c. 1.Firm one’s profit function: π 1 = (a-bQ 1 -bQ 2 )(Q 1 ) – cQ 1 = (a-c)Q 1 – bQ 1 2 -bQ 2 Q 1 . 2. d π 1 /dQ 1 = (a-c)-2bQ 1 -bQ 2 = 0. Q 1 = (a-c)/2b -0.5Q 2 . Reaction Function of Firm 1. Q 2 = (a-c)/2b -0.5Q 1 . Reaction Function of Firm 2. 3. Substituting Firm 2’s reaction function into Firm 1’s reaction function & solving for Q 1 : Q 1 = (a-c)/2b -0.5[(a-c)/2b -0.5Q 1 ] = (a-c)/4b + 0.25Q 1 . 0.75Q 1 = (a-c)/4b or Q 1 = (a-c)/3b = (1/3)[(a-c)/b]. 4. Market Quantity: Q= Q 1 +Q 2 = 2(a-c)/3b = (2/3)[(a-c)/b]. 5. Price : P = a – b[2(a-c)/3b] = a- (2/3)(a-c) = (1/3)a + (2/3)c = (1/3) [a + 2c]. II. Assumptions: n identical firms, linear demand and constant marginal cost Market Demand is P = a-bQ (where Q is market output) and MC = c. Q = Q i + Q j and Q j = (n-1)Q i j=2 j=2 1.Firm i’s profit function: π i = (P-MC)(Q i ) = (a-bQ i -b Q j – c)(Q i ) = aQ i -bQ i 2 -bQ i Q j – cQ i . j=2 j=2 2. d π i /dQ i = a-c -b2Q i -b Q j = 0 . j=2 by substitution: d π 1 /dQ i = (a-c) -b2Q i -b(n-1)Q i = 0 . = (a-c) –bQ i (2+n-1) = 0. = (a-c) –bQ i (n+1) = 0 rearranging, Q i = (a-c)/[b(1+n)]. 3. Market output: Q = nQ i = [n/(1+n)][(a-c)/b]. 4. Price: P = a-bQ = a-b{[n/(1+n)][(a-c)/b]} = a - {[n/(1+n)][(a-c)]} = [1/(1+n)] [a + nc]. Notice that as n Æ , then n/(1+n) Æ 1, and so P Æ c (perfect competition outcome). And if n = 1, then P = a – ½ (a-c) (monopoly outcome). III.Assumptions: 2 non-identical firms, Market Demand: P = a- Q = a- Q 1 -Q 2 (for simplicity we assume b=1) and MC 1 = c 1 and MC 2 = c 2 .
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